In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, the
topological structure of spinfoam or spin foam
[r-qc/0409061Introduction to Loop Quantum Gravity and Spin Foams">] consists of two-dimensional faces representing a configuration required by
functional integration
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differentia ...
to obtain a
Feynman's path integral
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional ...
description of
quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
. These structures are employed in
loop quantum gravity
Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attem ...
as a version of
quantum foam.
In loop quantum gravity
The
covariant formulation of loop quantum gravity provides the best formulation of the dynamics of the theory of
quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
– a
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
where the invariance under
diffeomorphisms of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
applies. The resulting path integral represents a sum over all the possible configurations of spin foam.
Spin network
A spin network is a one-dimensional
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
, together with labels on its vertices and edges which encode aspects of a spatial geometry.
A spin network is defined as a diagram like the
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
which makes a basis of
connections between the elements of a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
for the
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s defined over them, and for computations of amplitudes between two different
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
s of the
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. Any evolution of the spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network. A spin foam is analogous to
quantum history
In quantum mechanics, the consistent histories (also referred to as decoherent histories) approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural ...
.
Spacetime
Spin networks provide a language to describe the
quantum geometry of space. Spin foam does the same job for spacetime.
Spacetime can be defined as a superposition of spin foams, which is a generalized Feynman diagram where instead of a graph, a higher-dimensional complex is used. In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
this sort of space is called a 2-
complex. A spin foam is a particular type of
2-complex, with labels for
vertices, edges and
faces. The boundary of a spin foam is a spin network, just as in the theory of manifolds, where the boundary of an n-manifold is an (n-1)-manifold.
In Loop Quantum Gravity, the present Spin Foam Theory has been inspired by the work of
Ponzano–
Regge model. The concept of a spin foam, although not called that at the time, was introduced in the paper "A Step Toward Pregeometry I: Ponzano–Regge Spin Networks and the Origin of Spacetime Structure in Four Dimensions" by Norman J. LaFave. In this paper, the concept of creating sandwiches of 4-geometry (and local time scale) from spin networks is described, along with the connection of these spin 4-geometry sandwiches to form paths of spin networks connecting given spin network boundaries (spin foams). Quantization of the structure leads to a generalized Feynman path integral over connected paths of spin networks between spin network boundaries. This paper goes beyond much of the later work by showing how 4-geometry is already present in the seemingly three dimensional spin networks, how local time scales occur, and how the field equations and conservation laws are generated by simple consistency requirements. The idea was reintroduced in a 1997 paper and later developed into the
Barrett–Crane model
The Barrett–Crane model is a model in quantum gravity, first published in 1998, which was defined using the Plebanski action.
The B field in the action is supposed to be a so(3, 1)-valued 2-form, i.e. taking values in the Lie algebra of a spec ...
. The formulation that is used nowadays is commonly called EPRL after the names of the authors of a series of seminal papers,
but the theory has also seen fundamental contributions from the work of many others, such as
Laurent Freidel (FK model) and
Jerzy Lewandowski (KKL model).
Definition
The summary partition function for a spin foam model is